Compositions of series-parallel graphs. Brian Kell

Transcription

1 Compositions of series-parallel graphs Brian Kell March 29, 2006

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3 Abstract A composition of a graph is a partition of the vertex set such that the subgraph induced by each part is connected. In this paper we shall survey past results about compositions of graphs and present a new result which yields a linear-time method for computing the number of compositions of a series-parallel graph.

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5 Contents 1 Introduction 1 2 Definitions and notation Fundamental concepts in graph theory Compositions of graphs Standard families of graphs Past work Path graphs and complete graphs Counting principles for compositions Trees and almost complete graphs Cycle graphs Wheel graphs Ladder graphs Unions of graphs Series-parallel graphs Definitions of series-parallel graphs Applications to electric circuits Our results 45 6 Future work 53 i

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7 Acknowledgements I am very grateful to my research advisor, Dr. Jamie Radcliffe, without whose guidance and support over the past few months this thesis would never have been written. His enthusiasm for mathematics is contagious, and his insight has been invaluable. Dr. Jonathan Cutler has also been of much assistance. He has read several revisions of this thesis and given me helpful opinions and suggestions. I also wish to thank Dr. Judy Walker for suggesting that I make compositions of graphs my topic of research, for teaching my graph theory class this semester, and for encouraging me to apply for the Nebraska Mentoring through Critical Transition Points program, which has graciously funded my work. Further thanks go to Dr. Gordon Woodward, who among other things finally succeeded in persuading me to become a math major; Dr. Glenn Ledder, whose confidence in my work is much appreciated; Dr. Sohrab Asgarpoor, who provided answers about the electrical engineering implications of this topic; and Lori Mueller, who has helped to keep me on track and has taken care of many important administrative tasks. Finally, I must thank my parents, Tom and Patti Kell, for their unceasing support and encouragement, and for lending me their basement for a few days as I typed this paper. iii

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9 Chapter 1 Introduction A graph is a mathematical object consisting of a set of vertices, or points, and a set of edges connecting pairs of vertices. Typically graphs are drawn with dots representing the vertices and straight or curved lines representing the edges. For instance, Figure 1.1 shows a graph with seven vertices and nine edges. Figure 1.1: A graph. Graphs are the focus of the branch of mathematics known as graph theory. They are often used to model networks, such as electrical circuits, computer networks, or highway systems. Results from graph theory can be used to match jobs to qualified people, to design efficient methods for representing data in computers, to analyze chemical compounds, and to determine the optimal order in which to perform a series of interdependent tasks. As computers increase in importance, more applications of graph theory are appearing, as graphs provide a convenient mathematical model for real-world problems. One question that might be asked about a graph is how many different ways the graph can be cut up into pieces. There are several ways to interpret this question; the interpretation we shall examine here considers only the vertices which make up each part, and ignores the edges, except that the 1

10 Figure 1.2: Some compositions of the graph in Figure 1.1. Figure 1.3: Four cuttings that yield the same composition. 2

11 vertices in each part must be connected. Each such separation of the graph into pieces is called a composition of the graph. For example, the graph of Figure 1.1 has 130 compositions. Six of these compositions are shown in Figure 1.2. Note in particular that one composition contains all the edges of the graph; this is produced by not cutting any of the edges at all. Another composition is formed by cutting all the edges in the graph, making one piece for each vertex. An important point is that the four cuttings in Figure 1.3 are considered to be the same composition, since the pieces produced contain the same vertices. If G is an arbitrary graph, then, how many compositions does G have? This is an open question in graph theory. The answer is known for certain basic types of graphs, but no efficient technique for computing the number of compositions of a general graph is known. In this paper we consider a class of graphs called series-parallel graphs. Such a graph has two distinguished vertices, called the terminals. The simplest series-parallel graph has only two vertices and a single edge connecting them, as shown in Figure 1.4. (The terminals of a series-parallel graph will be drawn as unfilled circles to distinguish them from the other vertices.) Figure 1.4: The simplest series-parallel graph. More complicated series-parallel graphs may be constructed from simpler series-parallel graphs by joining them in series or in parallel. Joining graphs in series can be imagined as chaining the graphs together, connecting them at their terminals. For example, the graph in Figure 1.5 is formed by chaining together four copies of the simple graph of Figure 1.4. Figure 1.5: A graph formed by joining simpler graphs in series. When two or more graphs are joined in parallel, on the other hand, they are joined at both terminals. For example, the three graphs in Figure 1.6 can be joined in parallel to form the graph in Figure 1.7. Each of the graphs in Figure 1.6 are series-parallel graphs, so the graph in Figure 1.7 is also a series-parallel graph. This larger graph can itself be joined with another graph in series or in parallel. These graphs can of course become arbitrarily large. Figure 1.8 shows a series-parallel graph with 40 vertices and 53 edges. Not all graphs are series-parallel, however. 3

12 Figure 1.6: Three series-parallel graphs. Figure 1.7: The graphs of Figure 1.6 joined in parallel. The graph in Figure 1.9 is a common example of a graph that is not seriesparallel, because it cannot be constructed from simpler series-parallel graphs by joining them in series or in parallel. In the context of electrical networks, this graph is often called the Wheatstone bridge. 4

13 Figure 1.8: A larger series-parallel graph. Figure 1.9: A graph that is not series-parallel. 5

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15 Chapter 2 Definitions and notation In this chapter we shall set forth some definitions and notation which will be used throughout the rest of this paper. Many of these definitions are common graph theory terms; the reader may refer to Clark and Holton [6], Gould [10], Thulasiraman and Swamy [24], Wilson [25], or any number of other graph theory texts for further information. 2.1 Fundamental concepts in graph theory A graph G is an ordered pair (V, E) of sets. The set V, called the vertex set of G and alternately denoted by V (G), is a nonempty set of elements called the vertices of G. The set E, called the edge set of G and alternately denoted by E(G), is a possibly empty set of elements called the edges of G, such that each edge e in E(G) is an unordered pair {u, v} of vertices of G; these vertices are called the end vertices or endpoints of e. For example, consider again the graph of Figure 1.1. It has been redrawn in Figure 2.1 with the vertices labeled v 1 through v 7 and the edges labeled e 1 through e 9. The vertex set of this graph is {v 1, v 2, v 3, v 4, v 5, v 6, v 7 }, and the edge set is {e 1, e 2, e 3, e 4, e 5, e 6, e 7, e 8, e 9 }. Each edge has two endpoints; for instance, the endpoints of e 3 are v 1 and v 5. Note that this mathematical definition of a graph does not include information about how the graph is drawn; for many purposes, graph theory is concerned only with the interconnectedness of the vertices and the edges. In general, the endpoints of an edge need not be distinct. If the two endpoints of an edge are identical, then that edge is called a loop. Similarly, it is possible that two or more edges have the same endpoints; if this is the case, these edges are called parallel. For example, in the graph in Figure 2.2, 7

16 v 1 e 1 v 2 v 3 e 2 e 3 e4 e 5 e 6 e 7 v 4 v 5 v 6 e 8 e9 v 7 Figure 2.1: A graph with vertices and edges labeled. the edge e 1 is a loop, because the vertex v 1 serves as both of its endpoints, and the edges e 4 and e 5 are parallel, since the vertices v 2 and v 4 are the endpoints of both of these edges. In this paper, however, we shall consider only graphs which have no loops and no parallel edges. Such graphs are called simple. This is really not a major constraint, since loops can be discarded and sets of parallel edges replaced by a single edge (producing what is called the underlying simple graph), and the number of compositions will not be affected. We shall also restrict our scope to graphs with finitely many vertices and edges. If two vertices u and v are the endpoints of an edge e, we say that u and v are joined by e, that u and v are adjacent (written u adj v), and that e is incident on u and v. Two edges that share an endpoint are also called adjacent. The degree of a vertex v is the number of edges incident on v (counting loops twice, if the graph is not simple). It may be the case that two graphs have the same structure, differing only in how the vertices or edges are labeled or how the graph is drawn. In such a case we say that the two graphs are isomorphic. More precisely, a graph G is isomorphic to a graph H if there exists a one-to-one correspondence (a e 1 v 1 e 2 v 2 e 3 e 4 e 5 e 6 v 3 v 4 Figure 2.2: A graph with a loop and parallel edges. 8

17 u 1 u 2 u 3 u 4 v 1 v 2 w 1 w 2 w 3 w 4 v 3 v 4 u 5 u 6 v 5 v 6 u 7 u 8 v 7 v 8 w 5 w 6 w 7 w 8 Figure 2.3: Three mutually isomorphic graphs. bijection) between the vertices of G and the vertices of H, and a one-to-one correspondence between the edges of G and the edges of H, such that if e is an edge in G with endpoints u and v, then the corresponding edge f in H has as its endpoints the vertices w and x that correspond to u and v respectively. Such a pair of bijections is called a graph isomorphism. In this paper we shall not distinguish between isomorphic graphs, since if a graph G is isomorphic to a graph H, then G and H have the same number of compositions. To give an example, the three graphs in Figure 2.3 are mutually isomorphic. For each pair of these graphs, there are several different isomorphisms. The fact that the graph on the left and the graph in the center are isomorphic is readily apparent; one isomorphism between them is as follows. u 1 v 1, u 2 v 2, u 3 v 3, u 4 v 4, u 5 v 7, u 6 v 8, u 7 v 5, u 8 v 6. However, it is not so easy to see that the graph on the left is isomorphic to the graph on the right. One isomorphism between these two graphs is given below. u 1 w 5, u 5 w 6, u 2 w 3, u 3 w 1, u 4 w 7, u 6 w 4, u 7 w 2, u 8 w 8. An empty graph is one that has no edges. The singleton graph is the graph with one vertex and no edges. If G and H are graphs with V (H) V (G) and E(H) E(G), then H is said to be a subgraph of G. A subgraph H is called a proper subgraph of G if H G, that is, if V (H) V (G) or E(H) E(G), so that there exists 9

18 a vertex in G that does not exist in H, or an edge in G that does not exist in H. If U is a nonempty subset of the vertices of G, then the subgraph of G induced by U is the subgraph whose vertex set is U and whose edge set consists of all edges of G with both endpoints in U. Similarly, if F is a nonempty subset of the edges of G, then the subgraph of G induced by F is the subgraph whose edge set is F and whose vertex set consists of every vertex of G that is an endpoint of at least one edge in F. For instance, if we consider the graph of Figure 2.1, then the subgraph induced by the vertices {v 1, v 2, v 5, v 7 } is the subgraph shown on the left in Figure 2.4, and the subgraph induced by the edges {e 1, e 2, e 4, e 6, e 7 } is shown on the right. v 1 v 2 e 1 v 1 v 2 e 1 e 3 e 4 v 5 e 2 e 4 e 6 e 7 v 4 v 5 v 6 e 8 v 7 Figure 2.4: Subgraphs induced by a subset of vertices and a subset of edges. If e is an edge of G, then G e denotes the subgraph of G obtained by deleting the edge e, that is, the subgraph whose vertex set is V (G) and whose edge set is E(G) \ {e}. Analogously, if v is a vertex of G, then G v denotes the subgraph of G formed by deleting the vertex v and all edges incident on it, that is, the subgraph of G induced by V (G) \ {v}. Figure 2.5 shows two subgraphs of the graph G in Figure 2.1. The subgraph on the left is G e 9 ; that on the right is G v 5. Two useful operations we can perform on a graph are the identification of vertices and the contraction of edges. We identify two vertices u and v of a graph when we replace them by a new vertex w such that all edges that were previously incident on either u or v are now incident on w. (Any edges incident on both u and v become loops incident on w.) We contract an edge e of a graph when we delete it and then identify its endpoints. For instance, if we identify the vertices v 2 and v 5 in Figure 2.1, we obtain the graph illustrated on the left in Figure 2.6. Note that the edges e 1 and e 3 are now parallel, and e 4 has become a loop. If instead we contract the edge e 7, we produce the graph on the right. 10

19 v 1 e 1 v 2 v 3 v 1 e 1 v 2 v 3 e e 3 2 e4 e 5 e 6 e 7 v 4 v 5 v 6 e 8 e 2 e 5 v 4 v 6 e 9 v 7 v 7 Figure 2.5: Subgraphs produced by deleting an edge and by deleting a vertex. v 1 e 1 e 4 v 3 v 1 e 1 v 2 v 3 e 2 e 3 e 5 w e 6 e 7 v 4 e v 6 8 e 9 e 2 e 3 e 4 e 5 e 6 v 4 w e 8 e 9 v 7 v 7 Figure 2.6: Graphs produced by the identification of vertices and the contraction of an edge. 11

20 v 1 v 2 e 1 e 4 e 7 v 5 v 6 e 9 v 1 v 2 e 1 e 2 e 4 e 6 v 4 v 5 e 8 G v 7 H v 7 v 1 v 2 e 1 v 1 v 2 e 1 e 2 e 4 e 6 e 7 v 4 v 5 v 6 e 4 v 5 e 8 e9 G H v 7 G H v 7 Figure 2.7: The union and intersection of two graphs. The union of two graphs G and H, denoted G H, is the graph whose vertex set is V (G) V (H) and whose edge set is E(G) E(H). In the same way, the intersection of two graphs G and H with at least one vertex in common, written G H, is the graph whose vertex set is V (G) V (H) and whose edge set is E(G) E(H). (The requirement that G and H have at least one vertex in common is necessary since the vertex set of a graph must be nonempty.) Figure 2.7 shows two subgraphs of the graph in Figure 2.1, which are labeled G and H, and the union and intersection of these subgraphs. The union contains all vertices and edges that are in G or H, or both; the intersection contains only those vertices and edges that are in both G and H. Note that v 7 is an isolated vertex in the intersection. The join of two graphs G and H, denoted G + H, is the graph whose vertex set is V (G) V (H) and whose edge set is E(G) E(H) J, where J = { (v i, v j ) v i V (G) and v j V (H) }. In other words, the join of two graphs is their union together with edges that join every vertex in G to every vertex in H. Figure 2.8 illustrates the join of two graphs. 12

21 v 1 v 2 v 1 v 2 u 1 u 1 u 2 u 2 G v 3 v 4 H Figure 2.8: The join of two graphs. (a, w) v 3 v 4 G + H (a, y) a w y (b, w) (a, x) (b, y) (a, z) b c x z (c, w) (b, x) (c, x) (c, y) (b, z) G H G H (c, z) Figure 2.9: The Cartesian product of two graphs. The Cartesian product of two graphs G and H, which we shall denote by G H after Nešetřil [16], is the graph whose vertex set is V (G) V (H) and where two vertices (u, v) and (x, y) are adjacent if and only if (u = x) and (v adj y), or (u adj x) and (v = y). Figure 2.9 depicts the Cartesian product of two graphs. The reader may refer to Imrich and Klavžar [11] for a more thorough discussion of the Cartesian product and its properties. A finite sequence v 0 e 1 v 1 e 2 v 2... v k 1 e k v k is called a walk (from v 0 to v k ) in a graph G if for 0 i k, v i is a vertex in V (G), and for 1 i k, e i is an edge in E(G) with endpoints v i 1 and v i. 13

22 The vertex v 0 is called the origin of the walk, and the vertex v k is called the terminus. The vertices v 1, v 2,..., v k 1 are called the internal vertices of the walk. A walk is called closed or open according as v 0 = v k or v 0 v k. If all of the vertices v 0, v 1, v 2,..., v k of a walk are distinct, then the walk is called a path. It is easily seen that, given any two vertices u and v of a graph, any walk from u to v contains a path from u to v [6, Theorem 1.3]. For example, the heavy edges in Figure 2.10 form a walk from v 3 to v 4. There are several other walks from v 3 to v 4 in this graph. v 1 e 1 v 2 v 3 e 2 e 3 e4 e 5 e 6 e 7 v 4 v 5 v 6 e 8 e9 v 7 Figure 2.10: A path from v 3 to v 4. Two vertices u and v of a graph G are said to be connected if there exists a path in G from u to v. A graph is called connected if every two of its vertices are connected; otherwise it is called disconnected. A connected subgraph of a graph G is called a connected component of G if it is not a proper subgraph of any other connected subgraph of G. We shall denote the number of connected components of a graph G by k(g). For instance, the subgraphs G and H in Figure 2.7 are both connected, as is their union. However, their intersection is disconnected, since (to take an example) there does not exist a path from v 2 to v 7. The intersection has two connected components, so k(g H) = 2. If all of the edges e 1, e 2,..., e k of a walk are distinct, then the walk is called a trail. A nontrivial closed trail in which the origin and internal vertices are distinct is called a cycle. Figure 2.11 shows a graph with two cycles, which are drawn with heavy edges. A graph is called acyclic if it contains no cycles. A connected acyclic graph is called a tree. The three graphs in Figure 2.12 are trees. A bridge is an edge e of a graph G such that k(g e) > k(g). In other words, a bridge is an edge such that its deletion increases the number of connected components of the graph. In fact, if e is a bridge, then k(g e) = k(g) + 1; that is, its deletion increases the number of connected components 14

23 Figure 2.11: A graph with two cycles. Figure 2.12: Three trees. by exactly one [6, Theorem 2.6]. Similarly, a cut vertex is a vertex v of a graph G such that k(g v) > k(g). That is to say, a cut vertex is a vertex such that its deletion increases the number of connected components of the graph. For example, in the graph of Figure 2.1, the edge e 5 is a bridge, since if it were deleted the vertex v 3 would be isolated, forming two connected components. The vertex v 5 is a cut vertex, since its deletion disconnects the graph, as seen in the right-hand graph of Figure 2.5. The vertex v 6 is also a cut vertex in this graph. 2.2 Compositions of graphs The intuitive concept of cutting a graph into pieces has been formalized by Arnold Knopfmacher and M. E. Mays in [13]. A composition of a graph is a partition of the vertex set into one or more parts, such that the subgraph induced by each part is connected. This definition has several important implications. First, since a composition of a graph G is a partition of V (G), every vertex of G is in exactly one of the parts of the composition. In other words, a composition of G yields a set of connected subgraphs of G, and every vertex of G is in exactly one of these subgraphs. Second, as shown in Figure 1.3, not every set of connected subgraphs yields a distinct composition. In particular, when a composition is illustrated as a set of subgraphs, the edges exist solely to indicate the 15

24 v 1 v 2 v3 connected components; there may be other choices of edges that would imply the same composition. Third, in general the vertices of G cannot be arbitrarily partitioned, since the subgraph induced by each part must be connected. This means, for example, that the partition { {v1, v 5 }, {v 2, v 3, v 6 }, {v 4, v 7 } } of the vertices of the graph in Figure 2.1 is not a composition of the graph. This partition is illustrated in Figure 2.13; it is clear that not all of the subgraphs induced by each part are connected. v 4 v5 v 6 v 7 Figure 2.13: A partition of the vertices of a graph which does not yield a composition. We shall use the notation C(G), introduced in [13], to denote the number of compositions of a graph G. A composition of a graph is a partition of the vertex set, which can naturally be viewed as an equivalence relation. For a given composition C, then, we shall say that the vertices u and v are together in C (written u v) if u and v are in the same part. We shall sometimes refer to the parts of the composition, which are subsets of the vertex set of the graph, as the components of the composition. We shall say that an edge with endpoints u and v belongs to a particular composition if u v in that composition. 2.3 Standard families of graphs Certain graphs arise so often in graph theory that they have a standard notation. A complete graph is a simple graph in which every pair of vertices is joined by an edge. Any two complete graphs with the same number of vertices are isomorphic, so we refer to the complete graph on n vertices, 16

25 denoted K n. Figure 2.14 shows K 4, K 6, and K 10. Note that K 1 is the singleton graph. Figure 2.14: The complete graphs K 4, K 6, and K 10. If the vertices of a graph G can be partitioned into two nonempty subsets X and Y (that is, X Y = V (G) and X Y = ) in such a way that every edge of G joins a vertex in X to a vertex in Y, then G is said to be a bipartite graph, and the partition V (G) = X Y is called a bipartition of G. For example, the graph in Figure 2.3 is bipartite, which is most easily seen in the rightmost drawing; a bipartition of this graph is X = {w 1, w 2, w 3, w 4 }, Y = {w 5, w 6, w 7, w 8 }. A simple bipartite graph G with bipartition V (G) = X Y is called a complete bipartite graph if every vertex in X is joined to every vertex in Y. In this case, if X contains m vertices and Y contains n vertices, we denote this graph by K m,n. The graph in Figure 2.3 is not a complete bipartite graph, since (for example) there is no edge joining w 1 and w 8. Figure 2.15 shows the complete bipartite graph K 3,5. Note that K m,n is always isomorphic to K n,m. Figure 2.15: The complete bipartite graph K 3,5. A simple graph with vertex set {v 1, v 2,..., v n }, where v i is joined to v j if and only if i j = 1, is called a path graph. The path graph with n vertices is denoted P n. Figure 2.16 shows P 5. It is clear that P 1 is the singleton graph, and that P 2 is the same as the complete graph K 2 and the complete bipartite graph K 1,1. 17

26 Figure 2.16: The path graph P 5. A simple graph with vertex set {v 1, v 2,..., v n }, where v i is joined to v j if and only if i j ±1 (mod n), is called a cycle graph. The cycle graph with n vertices is denoted C n. Figure 2.17 illustrates C 4, C 6, and C 9. Note that by this definition C 1 is the singleton graph and C 2 is the path graph P 2. Additionally, C 3 is the same as the complete graph K 3. Figure 2.17: The cycle graphs C 4, C 6, and C 9. The complete bipartite graph K 1,n 1 is called the star graph with n vertices and is denoted S n. This name comes from the fact that such a graph is often drawn with one vertex in the center, surrounded by the other vertices, giving the appearance of a star. Figure 2.18 shows S 4, S 7, and S 12. By this definition S 2 and S 3 are the path graphs P 2 and P 3, respectively. Figure 2.18: The star graphs S 4, S 7, and S 12. The wheel graph with n vertices, denoted W n, can be formed from the cycle graph C n 1 by adding another vertex and joining this new vertex with every vertex in the original cycle graph; so W n is the join of C n and the singleton graph. By this definition, W 2 is the path graph P 2, W 3 is the cycle graph C 3, and W 4 is the complete graph K 4. For completeness, we shall take the wheel graph W 1 to be the singleton graph, as it is the only simple graph with exactly one vertex. Figure 2.19 illustrates W 4, W 5, and W 8. The vertices contributed by the cycle graph are called the outer vertices, 18

27 Figure 2.19: The wheel graphs W 4, W 5, and W 8. while the vertex contributed by the singleton graph is called the central vertex. The edges joining the central vertex to each of the outer vertices are called the spokes of the wheel graph. The grid graph G m,n is the Cartesian product P m P n of a path graph with m vertices and a path graph with n vertices. The grid graph G 4,6 is presented in Figure Clearly path graphs are a special case of grid graphs; the path graph P n is simply the grid graph G 1,n. Another special case is the class of ladder graphs. The ladder graph L n is defined to be the grid graph G 2,n. Figure 2.21 depicts the ladder graphs L 1, L 2, and L 5. Figure 2.20: The grid graph G 4,6. Figure 2.21: The ladder graphs L 1, L 2, and L 5. 19

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29 Chapter 3 Past work In this chapter we shall explore some previous work in the area of graph compositions. Knopfmacher and Mays focused on finding formulas for the number of compositions of various families of graphs, such as the wheel graphs and the complete bipartite graphs [13]. These results were extended by Ridley and Mays in [19], where they considered the unions of two graphs. 3.1 Path graphs and complete graphs The term composition was introduced by Knopfmacher and Mays in [13] and was motivated by the well-known combinatorial concept of a composition of a positive integer n, which is a representation of n as an ordered sum of positive integers [5]. (Some authors allow zeroes to appear in this sum [22]; the definition we use here may unambiguously be called a weak composition [1].) For example, several compositions of 7 are , , 6 + 1, and 7. (Note that and are distinct compositions of 7.) The relationship between compositions of graphs and compositions of positive integers becomes apparent when we consider the path graph P n. It is easy to see that any subset of the edges of P n yields a distinct composition. Since there are n 1 edges in P n, there are 2 n 1 subsets of these edges, and consequently we have the following theorem. Theorem 1. C(P n ) = 2 n 1. Knopfmacher and Mays arrive at this result by noting that any connected component of a composition of P n must itself be a path graph, and that the sum of the number of vertices in the components must of course be n. Therefore, a composition of P n yields a composition of n, and vice versa. 21

30 The number of compositions of n is 2 n 1 [1, Corollary 5.2]. This gives Theorem 1. If we consider the complete graph K n, we see that any partition of the vertices yields a composition; the subgraph induced by each part must necessarily be connected, since there is an edge between any two vertices in K n. Therefore we see that the number of compositions of K n is equal to the number of ways to partition n elements into nonempty subsets. This is exactly the definition of the Bell number B n, which gives the following result. Theorem 2. C(K n ) = B n. The series of Bell numbers B 1, B 2, B 3,... begins 1, 2, 5, 15, 52, 203, 877, 4140, , ,.... Much is known about this sequence; for further information, see [2] and [3]. In 1977 Gould compiled a bibliography of some 178 references on the Bell numbers [9]. The importance of the path graph P n and the complete graph K n lies in the following two theorems. Theorem 3. If G is a connected graph with n vertices, then it has at least C(P n ) = 2 n 1 compositions. We shall give a proof of Theorem 3 on page 25, after we have proved Theorem 7. Theorem 4. If G is a connected graph with n vertices, then it has at most C(K n ) = B n compositions. Proof. Any composition of G is a distinct partition of the vertex set of G. But there are only n elements in this set, and hence B n distinct partitions. Thus there can be at most B n compositions of G. Of course, this lower bound does not apply to disconnected graphs; the empty graph with n vertices has exactly one composition. 3.2 Counting principles for compositions In order to determine the number of compositions of a graph, it is necessary to know more than the number of vertices and edges in the graph. Knopfmacher and Mays give the example shown in Figure 3.1, in which both G 1 and G 2 have four vertices and four edges. It can be seen that C(G 1 ) = 10 but C(G 2 ) =

31 G 1 G 2 Figure 3.1: Two graphs with the same number of vertices and edges but different numbers of compositions. A basic theorem gives the number of compositions of a disconnected graph in terms of the number of compositions of each of its connected components. (It also provides a useful result in the case when a graph has a cut vertex.) Theorem 5. Let G be a graph that is the union of two subgraphs G 1 and G 2, that is, G = G 1 G 2. If there are no edges joining a vertex in G 1 \ G 2 to a vertex in G 2 \ G 1, and G 1 and G 2 have at most one vertex in common, then C(G) = C(G 1 ) C(G 2 ). Proof. This is an application of the multiplication principle [2, Section 3.1]; we form compositions of G by pairing compositions of G 1 with compositions of G 2 in all possible ways. A similarly useful result pertains to graphs with a bridge. Theorem 6. Let G be a graph, and suppose that the vertices of G can be partitioned into two nonempty, disjoint subsets V 1 and V 2 such that there exists exactly one edge e that joins a vertex in V 1 to a vertex in V 2. Let G 1 be the subgraph of G induced by V 1, and let G 2 be the subgraph of G induced by V 2. Then C(G) = 2 C(G 1 ) C(G 2 ). Proof. Let E be the subgraph of G consisting of the edge e with its two endpoints. First we see that G 1 and E have exactly one vertex in common, and so by Theorem 5 we have that C(G 1 E) = C(G 1 ) C(E). But clearly E is the path graph P 2, so C(E) = 2, and hence C(G 1 E) = 2 C(G 1 ). Now we note that G = G 1 E G 2 and that G 1 E and G 2 have exactly one vertex in common. Accordingly we apply Theorem 5 again, and find that C(G) = C(G 1 E G 2 ) = 2 C(G 1 ) C(G 2 ). The proof given by Knopfmacher and Mays uses the multiplication principle more directly. Let u be the endpoint of e in V 1, and let v be the other endpoint of e (so v V 2 ). We see that e is a bridge, since u and v are in the same connected component in G, but in different connected components in G e. For any composition of G 1 and any composition of G 2, there are 23

32 exactly two compositions of G: one in which e is included, combining the component containing u in G 1 and the component containing v in G 2, and one in which e is not included. This gives 2 C(G 1 ) C(G 2 ) compositions of G. Note that if a graph has a bridge, then the vertices of the graph can be partitioned in the manner described in Theorem 6; this is true even if the graph is disconnected. 3.3 Trees and almost complete graphs As a consequence of Theorem 6 and the fact that every edge in a tree is a bridge [6, Theorem 2.8], we have the following theorem. Theorem 7. Let T n be any tree with n vertices. Then C(T n ) = 2 n 1. Proof. The proof is by induction on n. Plainly T 1 is the singleton graph, which has exactly one composition, and 1 = 2 0. Now suppose that this result is true for all n less than some integer k, where k 2. Fix any edge e in T k ; this edge is a bridge, and so its deletion produces two connected components T 1 and T 2, each of which must be a tree. Suppose that T 1 has m vertices; then T 2 has k m vertices. Since m 1 and k m 1, we have that m < k and k m < k, so by our inductive hypothesis C(T 1 ) = 2 m 1 and C(T 2 ) = 2 k m 1. Because e is a bridge, Theorem 6 implies that C(T ) = 2 C(T 1 ) C(T 2 ) = 2 2 m 1 2 k m 1 = 2 k 1. The path graph P n is a tree with n vertices, so Theorem 1 is a special case of Theorem 7. Similarly, the star graph S n is a tree with n vertices, so C(S n ) = 2 n 1. Let T n be a tree with n vertices, as in Theorem 7. Then T n has 2 n 1 compositions, by Theorem 7; moreover it has exactly n 1 edges [6, Theorem 2.4]. Consider the power set of E(T n ), that is, the set of all subsets of E(T n ). There are 2 n 1 such subsets. Each composition of T n can be mapped to one of these subsets, namely, the set of edges which belong to T n, and plainly this mapping must be injective (one-to-one). But there are 2 n 1 compositions of T n and the same number of subsets of E(T n ), so this mapping must also be surjective (onto). Consequently we see that the power set of the edge set of a tree yields in a natural way the set of compositions of the tree. Using Theorem 7, we can now give a proof of Theorem 3, which states that a connected graph G with n vertices has at least C(P n ) = 2 n 1 compositions. 24

33 Proof of Theorem 3. Since G is connected, we can construct a spanning tree of G, which is a subgraph of G that contains all vertices of G and is a tree [6, Theorem 2.12]. This spanning tree has n vertices, and so by Theorem 7 it has 2 n 1 compositions; that is, there exist 2 n 1 partitions of the vertex set of the spanning tree which are compositions. But each of these compositions is of course a partition of the vertex set of G, and so we have at least 2 n 1 compositions of G. Knopfmacher and Mays also consider the case of a complete graph with one edge deleted, and arrive at the following result. Theorem 8. Let Kn denote the complete graph on n vertices with one edge deleted. Then C(Kn ) = B n B n 2. Proof. Let e be the deleted edge, and let u and v be the endpoints of e. The only compositions of K n which are not compositions of Kn are those in which one component consists only of u and v; in all other compositions of K n in which u and v are in the same component, there exists a third vertex w in that component, and the path from u to w to v still exists in Kn. Therefore, to count the number of compositions of Kn we must subtract from the number of compositions of K n all those compositions in which one component is {u, v}. By Theorem 2, there are B n compositions of K n. The vertices other than u and v form K n 2, of which there are B n 2 compositions. Hence, C(Kn ) = B n B n 2. The analysis becomes more complicated when more than one edge is deleted from a complete graph; the number of compositions of the resulting graph depends on whether the deleted edges are adjacent or not. The number of compositions of the complete graph K n with two edges deleted is shown in Table 3.1 for several values of n. n deleted edges adjacent deleted edges not adjacent Table 3.1: Number of compositions of the complete graph K n with two edges deleted. 3.4 Cycle graphs The number of compositions of the cycle graph C n can also be expressed with a simple formula, given in Theorem 9. 25

34 Theorem 9. C(C n ) = 2 n n. Proof. Choose any edge e and delete it. This produces the path graph P n. By Theorem 1, there are 2 n 1 compositions of P n. Each of these compositions is also a composition of C n. Moreover, for each of the compositions of P n, another composition of C n can be obtained by reinserting e, unless the composition of P n was formed by deleting no edges, or by deleting exactly one edge. In these cases, reinserting e results in the composition consisting of only one part, containing all n vertices. This composition is induced by n+1 different subsets of the edges: the entire edge set, and each of the n subsets containing all but one edge. Therefore, there are 2 2 n 1 n = 2 n n compositions of C n. Knopfmacher and Mays give in [13] a very brief sketch of a second proof of Theorem 9 which uses the concept of Lyndon compositions. A fuller exposition of this proof follows. A Lyndon composition of the positive integer n is an aperiodic composition that is lexicographically least among its cyclic permutations [13]. This means that , , and are not Lyndon compositions of 12, since the first two are periodic, and the last is not lexicographically least among its cyclic permutations (in fact, neither is the first). The cyclic permutations of the last composition are , , , , and , which arranged in lexicographic order are , , , , Therefore is a Lyndon composition of 12, since it is aperiodic and lexicographically least among its cyclic permutations. 26

35 Knopfmacher and Mays give, without proof, the number L(n) of Lyndon compositions of the integer n as L(n) = 1 ( n ) µ 2 d. (3.1) n d d n They also note that by this formula L(1) = 2. First, it is clear that the value L(1) = 2 is incorrect, since there is but a single composition of 1 (namely, 1 = 1). With the exception of this discrepancy, however, the above formula does give the correct number of Lyndon compositions of n. We shall now examine why this is so. The concept of Lyndon compositions is related to the concept of Lyndon words, which are aperiodic strings that are lexicographically least among their cyclic permutations [15]. This means that, for instance, the strings 0101 and are not Lyndon words, since they are periodic; and the string is not a Lyndon word, despite being aperiodic, because it is not lexicographically least among its cyclic permutations (however, its least cyclic permutation, 00101, is a Lyndon word). We may consider any binary string S of length n to be built from a unique Lyndon word, in the following sense: If S is aperiodic, then its corresponding Lyndon word is simply its lexicographically least cyclic permutation. Otherwise, S is periodic, and its corresponding Lyndon word is the lexicographically least cyclic permutation of the shortest repeating binary substring in S. To give an example, the string is periodic; its smallest repeating substring is The lexicographically least cyclic permutation of this substring is 0111, which is the Lyndon word from which the original string is built. A Lyndon word W of length d can be used to build a binary string S of length n only if d divides n, since the Lyndon word must repeat an integral number of times in S. Suppose that d does divide n. There are d distinct cyclic permutations of W, each of which, when repeated n/d times, will yield a distinct binary string of length n. Hence, the number of binary strings of length n is dl(d), d n where l(d) is the number of Lyndon words of length d. Since the number of binary strings of length n is 2 n, we see that 2 n = d n dl(d). 27

36 By the Möbius inversion formula [3, Section 12.7], we obtain l(n) = 1 n d n ( n ) µ 2 d. (3.2) d Interestingly, Equation 3.2 is identical to Equation 3.1. Therefore, it is claimed that there is a one-to-one correspondence between Lyndon words of length n and Lyndon compositions of n (except when n = 1, as noted above). Finding such a correspondence will prove Equation 3.1. Consider a row of n circles. A composition of n can be represented by separating this row of circles into one or more pieces. For example, the composition 11 = can be represented as shown in Figure 3.2. Figure 3.2: The composition 11 = Now consider the binary string , which is of length 11. We can express this string with circles, with representing 0 and representing 1, as shown in Figure 3.3. If we separate this row of circles after each 1 (that is, after each ), we obtain the composition 11 = , as seen in Figure Figure 3.3: The binary string Figure 3.4: The correspondence between and 11 = Note that the last digit of the binary string does not matter; the same composition could just as well have been represented by , as seen in Figure 3.5. Thus, for n 1, each each composition of n corresponds 28

37 Figure 3.5: The string also yields 11 = to exactly two binary strings of length n. This one-to-two correspondence occurs because the last digit of the binary string is irrelevant. When we consider Lyndon words, the situation is different. With the sole exception of the Lyndon word 0, the last digit of every Lyndon word is 1, for otherwise the string could be cyclically permuted to rotate the final 0 to the beginning, creating a lexicographically lesser string (because all Lyndon words other than 0 must contain the digit 1). This fact is convenient for us, since we can effectively ignore the last digit of Lyndon words when n 2. Since Lyndon words are aperiodic, their corresponding compositions are also aperiodic. Moreover, since Lyndon words are lexicographically least among their cyclic permutations, their corresponding compositions will be lexicographically greatest. This is because a Lyndon word will begin with as many 0s as possible, thus placing the largest part of the composition first. So we see that the compositions corresponding to Lyndon words are exactly those aperiodic compositions that are lexicographically greatest among their cyclic permutations. Each of these greatest compositions will correspond to exactly one Lyndon composition. Therefore, for n 2 there is a one-to-one correspondence between Lyndon words of length n and Lyndon compositions of n, thus providing a justification for Equation 3.1. In the case where n = 1, we have two Lyndon words (0 and 1), but only one Lyndon composition (1 = 1). This is the reason that Equation 3.1 yields L(1) = 2 instead of L(1) = 1. Now that we have established the validity of Equation 3.1, let us see how this can be used to prove Theorem 9. Let d be a divisor of n. To make a composition of the cycle graph C n, we may delete any n/d edges spaced regularly around C n, and, in each of the n/d segments of C n thus produced, delete edges so as to create a Lyndon composition of d, with the same Lyndon composition in each segment. (Note that there are d distinct ways in which one may delete n/d regularly spaced edges of C n.) Moreover, every such composition of C n can be produced in this way. The composition consisting solely of one part can be made by setting d = n and using the Lyndon composition d = d. All other compositions must have more than one part. If such a composition is aperiodic, then it 29

38 can be made by setting d = n and using the appropriate Lyndon composition of d. If instead it is periodic, let k be the number of times the smallest repeating unit repeats (called the primitive period of the composition [14]), set d = n/k, and choose the appropriate Lyndon composition of n/k. One minor problem remains, however, which is that the composition consisting of a single part is counted n times (once for the deletion of each edge of the graph, when d = n). So we must subtract n 1 from our final sum. Let L(d) denote the number of Lyndon compositions of d; so L(d) = L(d) for all d, except that L(1) = 1 but L(1) = 2. Then by the above argument we see that C(C n ) = dl(d) n + 1. d n Notice that because L(1) = L(1) + 1, but L(d) = L(d) for all other values of d, we may instead write C(C n ) = d n dl(d) n, which is indeed the formula given in [13]. By the Möbius inversion of Equation 3.1 we obtain 2 n = dl(d), d n so C(C n ) = 2 n n, which is Theorem 9. The sequence L(1), L(2), L(3),..., as given by Equation 3.1, begins 2, 1, 2, 3, 6, 9, 18, 30, 56, Wheel graphs If we take the wheel graph W 1 to be the singleton graph, W 2 to be the path graph P 2, and W 3 to be the cycle graph C 3, then the sequence {C(W n )} begins 1, 2, 5, 15, 43, 118, 316, 836, 2199, 5769, , , ,.... Lyndon compositions also arise in a formula for the number of compositions of the wheel graph W n. Here they are used as an index of summation. Note that in this formula C(P 0 ) is taken to be 1. 30

39 Theorem 10. For n > 1, C(W n ) = 2 n 1 n <d n 1 d a 1 + +a k =d where indicates a sum over Lyndon compositions of d. k C(P ai 1) (n 1)/d, Proof. We consider two cases. In the first case, the central vertex is together with no outer vertices. These compositions of the wheel graph correspond to compositions of the cycle graph C n 1, which by Theorem 9 has 2 n 1 (n 1) compositions. In the second case, the central vertex is together with one or more outer vertices. Consider the spokes that belong to such a composition. They separate the gaps between the outer vertices into one or more clusters. For example, if two adjacent spokes belong to the composition, then the gap between them is isolated from the other gaps. Since there are n 1 gaps in total, there may be a single cluster of gaps (if only one spoke belongs to the composition), or as many as n 1 clusters (if every spoke belongs to the composition). The arrangement of these gaps may be periodic (for example, if every second spoke belongs to the composition) or aperiodic. In either case, however, the arrangement corresponds to some Lyndon composition, possibly repeated an integral number of times. Since there are n 1 gaps in all, this must be a Lyndon composition of some integer d which divides n 1. This is the reason for the two summations: the first is a summation over all possible divisors of n 1, and the second is a summation over all Lyndon compositions of each divisor. There are d distinct cyclic permutations of this Lyndon composition, which will yield d distinct separations of the gaps; this gives the factor of d. For each cyclic permutation of this Lyndon composition, which must be repeated (n 1)/d times to go all the way around the outside of the wheel, the clusters of gaps contain within them clusters of outer vertices that are not together with the central vertex. If there are a i gaps in a cluster, then that cluster contains a i 1 such outer vertices. There are C(P ai 1) compositions of these outer vertices, and each cluster of vertices may be partitioned into a composition independently of the other clusters. This produces the product over the lengths a i of the clusters. The exponent of (n 1)/d exists because of the repetition of the Lyndon composition; the cluster of length a i will reappear (n 1)/d times around the wheel. Finally, note that if d = 1, then there is but one Lyndon composition (namely, 1 = 1). Since C(P 0 ) = 1 we get only one composition of the wheel 31 i=1

40 graph in this case (this is the composition in which every outer vertex is together with the central vertex). Therefore, we restrict d to be strictly greater than 1 in the first summation, and simply add 1 explicitly to the formula. This is why the formula begins 2 n 1 n rather than 2 n 1 n This is also the reason that the formula does not work for n = 1, since in the case of W 1 the composition where the central vertex is not together with any outer vertex is the same as the composition where the central vertex is together with all outer vertices. The value of C(W n ) is much easier to obtain using a recurrence relation noted by Knopfmacher and Mays, which is C(W 1 ) = 2, C(W 2 ) = 2, C(W n ) = 3 C(W n 1 ) C(W n 2 ) + n 2 for n 3. Of course, this value for C(W 1 ) is incorrect, for the reason noted in the proof. In [19], Ridley and Mays show that, for n 4, C(W n ) = F 2n 1 + F 2n 3 n + 1 = L 2n 2 n + 1, where F k denotes the kth Fibonacci number and L k denotes the kth Lucas number. 3.6 Ladder graphs The ladder graph L 1 is simply the path graph P 2, so C(L 1 ) = 2 by Theorem 1. The ladder graph L 2 is the cycle graph C 4, so C(L 2 ) = 12 by Theorem 9. Further values of C(L n ) can be described by a recurrence relation. Theorem 11. The number of compositions of the ladder graph L n satisfies the recurrence relation C(L 1 ) = 2, C(L 2 ) = 12, C(L n ) = 6 C(L n 1 ) + C(L n 2 ) for n 3. Proof. Denote the vertices of the ladder graph L k by v 1,1, v 1,2, v 2,1, v 2,2,..., v k,1, v k,2. We shall call v k,1 and v k,2 the tail vertices of L k. If the tail 32

41 L n 1 (1) v n 1,1 v n,1 v n 1,2 v n,1 (5) (2) (6) (3) (7) (4) (8) Figure 3.6: The eight cases considered in the proof of Theorem 11. vertices are together in a composition of a ladder graph, we shall say that the composition is united; otherwise, we shall say that the composition is divided. Let D k represent the number of divided compositions of L k, and let U k represent the number of united compositions. To make L n from L n 1, we add two new tail vertices v n,1 and v n,2, and three new edges which join v n,1 to v n 1,1, v n,2 to v n 1,2, and v n,1 to v n,2. Any combination of these three edges can belong to a composition of L n. This results in eight cases to consider, as seen in Figure 3.6. Suppose we wish to make a divided composition of L n. For any divided composition of L n 1 in which the tail vertices are not together, cases (1), (2), (5), and (6) will result in a divided composition of L n. If we start with a united composition of L n 1, then cases (1), (2), and (5) will yield a divided composition of L n. Therefore D n = 4 D n U n 1. On the other hand, suppose we wish to construct a united composition of L n. From a divided composition of L n 1, cases (3), (4), and (7) yield distinct united compositions of L n ; case (8) would make the composition of L n united. Starting with a united composition of L n 1, we can make a united composition of L n via case (3) or case (8); cases (4) and (7) yield the same composition as case (8). Hence U n = 3 D n U n 1. Thus, we have C(L n ) = D n + U n = 7 D n U n 1. 33

42 But we also have D n 1 U n 1 = D n 2 + U n 2 = C(L n 2 ). Consequently, C(L n ) = 6(D n 1 + U n 1 ) + (D n 1 U n 1 ) = 6 C(L n 1 ) + C(L n 2 ). Knopfmacher and Mays observe that this recurrence relation is the same as that for the denominators in the continued fraction expansion of Unions of graphs A general result for the number of compositions of a graph expressed as a union of two subgraphs is given by Ridley and Mays in [19]. They consider a graph expressed as a union of edge-disjoint graphs, but in fact their results hold even if the subgraphs have one or more edges in common. Let G be a graph expressed as a union G = A B, and let C be the equivalence relation defined by some composition C of G. If we restrict the application of C to the vertices of A, the equivalence classes so produced do not necessarily make a composition of A, since one or more of them may not be connected when we use edges from A only. However, we do obtain a unique composition C A of A defined by all edges in A with endpoints u and v such that u C v, that is, all edges in A that belong to C. We obtain a similar composition C B for B. We shall say that C induces the pair (C A, C B ). (Ridley and Mays say that C is valid for C A and C B if C induces C A and C B.) The procedure for counting distinct compositions of G, then, is to consider the pairing of each composition C A of A with each composition C B of B, and to count the resulting composition of G only if it induces C A and C B. Since each composition of G induces exactly one pair (C A, C B ), this procedure will count each composition of G exactly once. For a given composition C of a graph G, we shall classify each pair {u, v} of vertices in G into one of three categories. If u v, that is, if u and v are in the same component of C, then we shall say that u and v are together in C. If u and v are not together in C, but there exist vertices x and y adjacent in G such that u and x are together and v and y are together, then we shall say that u and v are nearby in C. If u and v are neither together nor nearby in C, we shall say that they are distant in C. Ridley and Mays use the terms type (0), type (1), and type (2) to refer to pairs of vertices that are together, 34